Answer Happy

1. The jump conditions for inviscid flow with piecewise linear velocity profiles, and piecewise constant density profiles, are Aloºd Lo”.] =0, a lo{U – elu – Uv- 0-100 2.] = 0 UoY Δ = 0, gu U cu - U'r = , (1) where A indicates the change in a quantity across a discontinuity in the basic flow evaluated on either side of the discontinuity, v = v(y) is the vertical component of disturbance velocity, c is the disturbance phase velocity, U = U(y) is the basic velocity, P = P(y) is the basic density, and g is acceleration due to gravity. Consider the uniform shear flow U= h (2) where u, and h are constants. (a) Show that for (2), the jump conditions, (1), can be simplified to dgho Dy=vi, tv + (3) (Uoyo - ch) when d<1, u for y > 36 12 for y < yo (4) where all quantities in (3) are evaluated at y = yo. You need to make the Boussinesq approximation, i.e. assume that the buoyancy term, involving 9, is 0(1), but that other O(d) terms are negligible. (b) Use (3) to show 30/0 dg (5) h 2a when the flow (2) is unbounded, and the density profile is given in (4), where a is the wavenumber p= {Po(1 + d) for you y> 9 C (c) Consider the density profile P= PO for y>h po(1+d) for-h<ych, d < 1. po(1+d)? for y <h (6)
(c) Consider the density profile Po for y>h P = { po(1 + d) for-h<y<h, d<1. (6) po(1+ d) for y<-h (i) Why can jump conditions in the form of (3) be used at both y=h and y = -h? -- (ii) Show that the dispersion relation for (2) and (6), for unbounded flow, using jump conditions in the form of (3), can be written 46?' – 4a (2a + J)* + 4ālā - J) + (1 - 1)/2 = 0, (7) when expressed in terms of the dimensionless variables dgh a = ah, J = (8) ul C UO (iii) Show that in the short-wave limit, ã » 1, (7) can be factorized with solutions corresponding to those of (5). Explain this result, (iv) What is the condition on J for instability when a < 1 and 7, J = 0(1)? Hence, find a more accurate condition for instability when a < 1, c = (a/) and J = 0(1). (50 marks)